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Vol. 4 (1999) Paper no. 10, pages 1–13.Journal URL
http://www.math.washington.edu/~ejpecp/
Paper URL
http://www.math.washington.edu/~ejpecp/EjpVol4/paper10.abs.html
THICK POINTS FOR TRANSIENT SYMMETRIC STABLE PROCESSES
Amir Dembo
Department of Mathematics and Department of Statistics Stanford University
Stanford, CA 94305 amir@stat.stanford.edu
Yuval Peres
Mathematics Institute, Hebrew University, Jerusalem, Israel and Department of Statistics, University of California
Berkeley, CA 94720 peres@math.huji.ac.il
Jay Rosen
Department of Mathematics College of Staten Island, CUNY
Staten Island, NY 10314 jrosen3@idt.net
Ofer Zeitouni
Department of Electrical Engineering Technion
Haifa 32000, Israel zeitouni@ee.technion.ac.il
AbstractLetT(x, r) denote the total occupation measure of the ball of radiusr centered at x for a transient symmetric stable processes of indexβ < dinIRdand Λβ,ddenote the norm of the
convolution with its 0-potential density, considered as an operator onL2(B(0,1), dx). We prove
that sup|x|≤1T(x, r)/(rβ|logr|)→βΛβ,da.s. asr→0. Furthermore, for anya∈(0, βΛβ,d), the
Hausdorff dimension of the set of “thick points”x for which lim supr→0T(x, r)/(rβ|logr|) =a,
is almost surelyβ−a/Λβ,d; this is the correct scaling to obtain a nondegenerate “multifractal
spectrum” for transient stable occupation measure. We also show that the lim inf scaling of T(x, r) is quite different: we exhibit positive, finite, non-random c′
β,d, Cβ,d, such that c′β,d <
supxlim infr→0T(x, r)/rβ< Cβ,d a.s.
KeywordsStable process, occupation measure, multifractal spectrum
AMS Subject Classification60J55
This research was supported, in part, by grants from the National Science Foundation, the U.S.-Israel Binational Science Foundation, MSRI and PSC-CUNY.
Thick Points for Transient Symmetric Stable Processes
Amir Dembo
∗Yuval Peres
†Jay Rosen
‡Ofer Zeitouni
§Abstract
LetT(x, r) denote the total occupation measure of the ball of radiusrcentered atxfor a transient symmetric stable processes of index β < d inIRd and Λβ,d denote the norm of
the convolution with its 0-potential density, considered as an operator onL2
(B(0,1), dx). We prove that sup|x|≤1T(x, r)/(r
β|logr|) → βΛ
β,d a.s. as r → 0. Furthermore, for
any a ∈ (0, βΛβ,d), the Hausdorff dimension of the set of “thick points” x for which
lim supr→0T(x, r)/(rβ|logr|) = a, is almost surely β−a/Λβ,d; this is the correct scaling
to obtain a nondegenerate “multifractal spectrum” for transient stable occupation measure. We also show that the lim inf scaling ofT(x, r) is quite different: we exhibit positive, finite, non-randomc′
β,d, Cβ,d, such thatc′β,d<supxlim infr→0T(x, r)/rβ< Cβ,d a.s.
1
Introduction
The symmetric stable process{Xt} of index β < d in IRd does not hit fixed points, hence does
not have local times. Nevertheless, the path will be “thick” at certain points in the sense of having larger than “usual” occupation measure in the neighborhood of such points. Our first result tells us just how “thick” a point can be.
Recall that theoccupation measureµX
T is defined as
µXT (A) =
Z T
0 1A(Xt)dt
for all Borel setsA⊆IRd. As usual, we let
u0(x) = cβ,d
|x|d−β (1.1)
denote the 0-potential density for {Xt}, where cβ,d = 2−βπ−d/2Γ(d−2β)/Γ(β2). Let Λβ,d denote
the norm of
Kβ,df(x) =
Z
B(0,1)u
0(x−y)f(y)dy
considered as an operator fromL2(B(0,1), dx) to itself. Throughout, B(x, r) denotes the ball inIRd of radiusr centered atx.
∗Research partially supported by NSF grant #DMS-9403553.
†Research partially supported by NSF grants #DMS-9404391 and #DMS-9803597 ‡Research supported, in part, by grants from the NSF and from PSC-CUNY. §
Theorem 1.1 Let {Xt} be a symmetric stable process of index β < d in IRd. Then, for any
R∈(0,∞) and any T ∈(0,∞],
lim
ǫ→0|xsup|≤R
µXT(B(x, ǫ))
ǫβ|logǫ| =βΛβ,d a.s. (1.2)
Remarks:
• Our proof shows that for anyT ∈(0,∞],
lim
ǫ→00≤supt≤T
µXT(B(Xt, ǫ))
ǫβ|logǫ| =βΛβ,d a.s. (1.3)
• The scaling behavior of stable occupation measure around any fixed timetis governed by the stable analogue of the LIL of Ciesielski-Taylor, see [11]; for anyT ∈(0,∞] andt≤T,
lim sup
ǫ→0
µXT(B(Xt, ǫ))
ǫβlog|logǫ| =
βΛβ,d
2 a.s. (1.4)
(1.2) and (1.3) are our analogues of L´evy’s uniform modulus of continuity. The proof of such results for Brownian occupation measure was posed as a problem by Taylor in 1974 (see [12, Pg. 201]) and solved by us in [3, Theorem 1.2].
Our next result is related to (1.4) in the same way that the formula of Orey and Taylor [5] for the dimension of Brownian fast points is related to the usual LIL. It describes the multifractal nature, in a fine scale, of “thick points” for the occupation measure of {Xt} (We call a point
x ∈ IRd a thick point if x is in the set considered in (1.5) for some a > 0; similiarly,t > 0 is called athick timeif it is in the set Thicka considered in (1.6) for somea >0 andT >0.)
Theorem 1.2 Let {Xt} be a symmetric stable process of index β < d in IRd. Then, for any
T ∈(0,∞]and all a∈(0, βΛβ,d],
dim{x∈IRd
lim sup
ǫ→0
µXT(B(x, ǫ))
ǫβ|logǫ| =a}=β−aΛ −1
β,d a.s. (1.5)
Equivalently, for anyT ∈(0,∞]and all a∈(0, βΛβ,d],
dim{0≤t < T
lim sup
ǫ→0
µX
T (B(Xt, ǫ))
ǫβ|logǫ| =a}= 1−aΛ −1
β,d/β a.s. (1.6)
Denote the set in (1.6) byThicka. Then Thicka 6=∅ at the critical value a=βΛβ,d.
For all a∈ (0, βΛβ,d), the union Thick≥a := ∪b≥aThickb has the same Hausdorff dimension as Thicka a.s., but its packing dimension a.s. satisfiesdim
P(Thick≥a) = 1. Equivalently,
dimP{x∈IRd
lim sup
ǫ→0
µXT(B(x, ǫ))
ǫβ|logǫ| ≥a}=β a.s. (1.7)
• Combining (1.5) and (1.2) we see that
sup
x∈IRd
lim sup
ǫ→0
µX
∞(B(x, ǫ))
ǫβ|logǫ| =βΛβ,d a.s.
In particular, the sets in (1.5) and (1.6) are a.s. empty for anya > βΛβ,d,T ∈(0,∞].
• For anyx /∈ {Xt
0≤t≤T}andǫsmall enough, µ
X
T (B(x, ǫ)) = 0. Hence, the equivalence
of (1.5) and (1.6) is a direct consequence of the equivalence between spatial and temporal Hausdorff dimensions for stable motion due to Perkins-Taylor (see [7, Eqn. 0.1]), together with the fact that{Xt
0≤t≤T} − {Xt
0≤t≤T} is countable.
Our next theorem gives a precise estimate of the total duration in [0,1] that the stable motion spends in balls of radiusǫthat have unusually high occupation measure. Such an estimate (which is an analogue in our setting of the “coarse multifractal spectrum”, cf. Riedi [10]), cannot be inferred from Theorem 1.2.
Theorem 1.3 Let {Xt} be a symmetric stable process of index β < d in IRd, and denote
Lebesgue measure onIR byLeb. Then, for any a∈(0, βΛβ,d),
lim
ǫ→0
logLebn0≤t≤1 µ
X
1 (B(Xt, ǫ))≥aǫβ|logǫ|
o
logǫ =aΛ
−1
β,d a.s.
Theorem 1.3 will be derived as a corollary of the following theorem which provides a pathwise asymptotic formula for the moment generating function of the ratioµX1 (B(Xt, ǫ))/ǫβ.
Theorem 1.4 Let {Xt} be a symmetric stable process of index β < d in IRd. Then, for each
θ <Λ−β,d1,
lim
ǫ→0
Z ∞
0
eθµX1 (B(Xt,ǫ))/ǫβdt=
IEeθµX∞(B(0,1))2 a.s. (1.8)
It follows from the proof that both sides of (1.8) are infinite ifθ≥Λ−β,d1.
The thick points considered thus far are centers of ballsB(x, ǫ) with unusually large occupation measure for infinitely many radii, but these radii might be quite rare. The next theorem shows that for “consistently thick points” where the balls B(x, ǫ) have unusually large occupation measure forallsmall radiiǫand the same centerx, what constitutes “unusually large” must be interpreted more modestly.
Theorem 1.5 Let {Xt} be a symmetric stable process of index β < d in IRd. Then for some
non-random 0< c′β,d< Cβ,d<∞ we have,
c′β,d≤ sup
x∈IRd
lim inf
ǫ→0
µX∞(B(x, ǫ))
ǫβ ≤Cβ,d a.s. (1.9)
• In particular, replacing the lim sup by lim inf in (1.5) and (1.6) results with an a.s. empty set for alla >0.
• The new assertion in (1.9) is the right hand inequality; the inequality on the left is an immediate consequence of the existence of “stable slow points”, see [6, Theorem 22].
It is an open problem to determine the precise asymptotics in (1.9), even in the special case of Brownian motion.
This paper generalizes the results of our paper [3] on thick points of spatial Brownian motion to all transient symmetric stable processes. There are several sources of difficulty in this extension:
• Ciesielski-Taylor [2] provide precise estimates for the tail of the Brownian occupation measure of a ball in Rd; the existing estimates for stable occupation measure are not as
precise.
• The L´evy modulus of continuity for Brownian motion was used in [3] to obtain long time intervals where the process does not exit certain balls.
We overcome these difficulties by
• A spectral analysis of the convolution operator defined by the potential density.
• Conditioning on the absence of large jumps in certain time intervals; this creates strong dependence between different scales, and our general results on “random fractals of limsup type” are designed to handle that dependence.
Sections 2 and 3 state and prove the two new ingredients which are needed in order to establish the results in the generality stated here, that is the Localization Lemma 2.2 and the Exponen-tial Integrability Lemma 3.1. Applying these lemmas, Section 4 goes over the adaptations of the proofs of [3] which allows us to establish the results stated above for all transient stable occupation measures.
2
Localization for stable occupation measures
We start by providing a convenient representation of the law of the total occupation measure µX
∞(B(0,1)). This representation is the counterpart of the Ciesielski-Taylor representation for
the total occupation measure of spatial Brownian motion in [2, Theorem 1].
Lemma 2.1 Let {Xt} be a symmetric stable process of index β < d in IRd. Then, for any
u >0,
PµX∞(B(0,1))> u=
∞
X
j=1
ψje−u/λj , (2.1)
where λ1 > λ2 ≥ · · · ≥ λj ≥ · · ·> 0 are the eigenvalues of the operator Kβ,d with the
corre-sponding orthonormal eigenvectors φj(y), ψj :=φj(0)(1, φj)B(0,1) and the infinite sum in (2.1)
Proof of Lemma 2.1: We use pt(x) to denote the transition density of {Xt} and for ease
of exposition write K,Λ for Kβ,d,Λβ,d respectively, and let J = µX∞(B(0,1)). While u0 is in
general only inL1(B(0,1), dx), each application of K lowers the degree of divergence byβ (this is easily seen by scaling), sov:=Km−1u0, the convolution kernel ofKm is bounded and in fact continuous form large enough. Fix suchm and note that for anyn≥m,
IE (Jn) = IE
n=mzn−mun/n!, by the standard Neumann series for the resolvent, for
anyz∈IC such that|z|<Λ−1 :=θ∗,
SinceK is a convolution operator onB(0,1) with locallyL1IRd, dxkernel, it follows easily as in [4, Corollary 12.3] thatKis a (symmetric) compact operator. Moreover, the Fourier transform relationR
ei(x·p)u0(x)dx=|p|−β >0 implies thatKis strictly positive definite. By the standard
theory for symmetric compact operators,K has discrete spectrum (except for a possible accu-mulation point at 0) with all eigenvalues positive, of finite multiplicity, and the corresponding eigenvectors of K, denoted{φj} form a complete orthonormal basis of L2(B(0,1), dx) (see [8,
Theorems VI.15, VI.16]). Moreover, (f, Kg)B(0,1)>0 for any non-negative, non-zero,f, g, so by the generalized Perron-Frobenius Theorem, see [9, Theorem XIII.43], the eigenspace correspond-ing to Λ =λ1 is one dimensional, and we may and shall chooseφ1 such thatφ1(y) >0 for all
y∈B(0,1). Noting thatφj are also eigenvectors of (I−zK)−1 with corresponding eigenvalues
where cj := (1, φj)(v, φj) is absolutely summable. Since both integrals in (2.5) are analytic in
the strip Rez ∈ [−3θ∗/4,−θ∗/4], and agree for z real inside the strip, they agree throughout
this strip. Consideringz=−θ∗/2 +it,t∈IR we have that
fm(u) = ∞
X
j=1
cjλ−j1e−u/λj (2.6)
a.e. onu ≥0 by Fourier inversion and hence for all u >0 by right continuity. Considering the (m−1)-st derivative of (2.6), the uniform convergence ofP
jcjλ−jke−u/λj fork= 1, . . . , m and
u away from 0, shows that
P(J > u) =
∞
X
j=1
cjλ−jme−u/λj
for all u > 0. Recalling that v, the convolution kernel of Km, is bounded and continuous, we have thatλmj φj = Kmφj are continuous and bounded functions, andKmφj(0) = (v, φj)B(0,1).
Therefore φj(0) = λ−jm(v, φj)B(0,1) and thus cjλ−jm = (1, φj)φj(0) = ψj for allj, establishing
(2.1). 2
With the aid of (2.1) we next provide a localization result for the occupation measure of{Xt}.
Lemma 2.2 (The Localization Lemma) Let {Xt} be a symmetric stable process of index
β < d in IRd. Then, with θ∗ = Λ−β,d1, for some c0, c1 < ∞, t ≥ c0ud/(d−β), and all u > 0
sufficiently large
c−11e−θ∗u≤PµXt (B(0,1))≥u≤PµX∞(B(0,1))≥u≤c1e−θ
∗u
(2.7)
Withh(ǫ) =ǫβ|logǫ| and anyρ > d/(d−β), consideringu=a|logǫ|in (2.7), by stable scaling
we establish that for anya >0 and ǫ >0 small enough
c−11ǫaθ∗ ≤PµXǫβ|logǫ|ρ(B(0, ǫ))≥ah(ǫ)
≤PµX∞(B(0, ǫ))≥ah(ǫ)≤c1ǫaθ
∗
(2.8)
Proof of Lemma 2.2: LetJt:=µXt (B(0,1)). Recall thatφ1 is a strictly positive function on
B(0,1), hence in (2.1) we have ψ1>0 and θ∗ =λ−11 < λ−21, implying that
lim
u→∞P(J∞> u)e θ∗u
=ψ1∈(0,∞) (2.9)
out of which the upper bound of (2.7) immediately follows.
Turning to prove the corresponding lower bound, letτz := inf{s:|Xs|> z}, noting that by [1,
Proposition VIII.3] for some c >0 and anyu >0,z >1,t >0,
P(Jt> u)≥P(Jτz > u)−P(τz > t)≥P(Jτz > u)−c
−1exp(−ctz−β). (2.10)
LetJ and J′ denote two independent copies ofJ
∞. Noting that Py(J′ > u)≤P(J′ > u) for
anyy∈IRd,u >0, and using the strong Markov property, it is not hard to verify that
P(J∞> u)≤P(Jτz > u) +P(J +J
′> u) sup
|v|>z
Pv(inf
(c.f. [3, (3.6) and (3.7)] where this is obtained for the Brownian motion). Recall that
Pv(inf
s≥0|Xs|<1)≤c(β, d)|v|
−(d−β)∧1
(see [11, Lemma 4]). By (2.9) it follows that for someC <∞ and allu >0,
P(J +J′ > u)≤C(1 +u)e−θ∗u (2.12)
(c.f. [3, (3.8)] for the derivation of a similar result). Hence, for some C1,C2, c0 large enough,
takingz =zu :=C1u1/(d−β)and tu :=C2uzβu =c0ud/(d−β) one gets from (2.10) and (2.11) that
for somec′ >0 and allt≥t u
P(Jt> u)≥c′e−θ
∗u
(2.13)
as needed to complete the proof of the lemma. 2
3
Exponential integrability
Lemma 3.1 Let {Xt} denote the symmetric stable process of index β in IRd with β < d and
ψ(x) :=|x|−β1{|x|≤1}. Then, for anyθ∈(0, d−β) and
λ <Λ−β,d1(θ) := 2βΓ(
d−θ 2 )Γ(
β+θ 2 )
Γ(d−θ2−β)Γ(θ2), (3.1)
there exists κλ,θ <∞ such that for all |y| ≤1
IEy
exp(λ
Z ∞
0 ψ(Xt)dt)
≤κλ,θ|y|−θ. (3.2)
Proof of Lemma 3.1: As before,u0(x) =c
β,d|x|β−d denotes the 0-potential density for{Xt}.
Fixingθ∈(0, d−β) let Λβ,d(θ) denote the norm of
Kf(y) =|y|θ
Z
B(0,1)
u0(y−x)|x|−(β+θ)f(x)dx
considered as an operator fromL∞(B(0,1), dx) to itself. Recall the Fourier transform relation
R
ei(x·p)u0(x)dx=|p|−β, implying that
Λβ,d(θ) = sup y∈B(0,1)
(K1)(y) = sup
y
Z
IRd
cβ,d|y|θdx
|y−x|d−β|x|β+θ =
cd−θ,d
cd−θ−β,d
. (3.3)
Since cα,d= 2−απ−d/2Γ(d−2α)/Γ(α2) for any α∈(0, d), we obtain that
Λβ,d(θ) = 2−β
Γ(d−θ2−β)Γ(θ2)
Γ(d−2θ)Γ(β+θ2 ) , (3.4)
as stated in the lemma. Forg(y) =|y|θ andλ∈(0,Λ
β,d(θ)−1), the series
G(y) =
∞
X
n=0
converges uniformly inL∞(B(0,1), dx). It is easy to check that for all n,
resulting with the bound (3.2). 2
4
Proofs
Throughout we writeθ∗ for Λ−β,d1,h(ǫ) forǫβ|logǫ|and takeρ(ǫ) :=|logǫ|ρfor someρ > d/(d−β) as in the localization bound of (2.8).
Proof of Theorem 1.2, lower bound: In proving the lower bound it suffices to assume that T is finite; by stable scaling, it is enough to consider T = 2 or equivalently, −1 ≤ t ≤ 1 in (1.6). Our proof follows closely the proof of [3, Corollary 4.1] to which the reader is referred for notation and details. Take ǫn = n32−n/β, n = 1,2, . . . and bǫn = 1− |logǫn|
stable scaling and [1, Proposition VIII.4] it is easily verified that the second condition in (4.1) has probability at least 1/2 for n sufficiently large. By stable scaling, the lower bound of (2.8) directly implies that for allI ∈ Dn and nsufficiently large
P
Noting that the two conditions in (4.1) are independent, we see that forI ∈ Dn, and allnlarge
enough,
pn:=P(ZI= 1)≥2−aθ
∗n/β
. (4.2)
We will now apply [3, Theorem 2.1] with gauge function
The lower bound on the Hausdorff dimension in Theorem 1.2 now follows as in the proof of [3, Corollary 4.1], with the results about Packing dimension obtained by applying [3, Corollary 2.4]. The fact thatThicka6=∅ at the critical value a=βΛβ,dfollows by the same argument as in [3,
Section 4], using here the dense open sets
Thick(a, h) := [ ǫ,ǫ′∈(0,h)
n
0< t < T
µXT (B(Xt, ǫ))
ǫβ|logǫ| > aand
µXT (B(Xt−, ǫ′)) ǫ′β|logǫ′| > a
o
,
where we used the fact that the process{Xt} is right-continuous with left limits and has only a
countable set of jumps.
Proof of Theorem 1.2, upper bound: This follows as in [3, Section 5] where the bound
PµX∞(B(0,(1 +δ)ǫ))≥(1−2δ)ah(ǫ)≤cǫ(1−4δ)aθ∗ (4.3)
follows from (2.8), and for the standard estimate for stable hitting probabilities
P(σǫ(x)<∞)≤c(β, d)(
ǫ |x|)
d−β∧1, (4.4)
withσǫ(x) := inf{t≥0 :Xt∈B(x, ǫ)}, see [11, Lemma 4].
Proof of Theorem 1.1: With the above estimates, as well as the lower bound (2.8) of the Localization Lemma 2.2, this follows directly as in the corresponding proof of [3, (1.7)], using nowδǫ =ǫβρ(ǫ).
Proof of Theorem 1.4: In the course of proving Lemma 2.1 we verified among other things that IE(exp(θµX∞(B(0,1))) < ∞ for all θ < θ∗ (see (2.3)). While u0 ∈/ L2(B(0,1), dx) for β ≤d/2, we have thatKiu0 ∈L2(B(0,1), dx) for all ilarge enough, which is all that one needs
when extending [3, Lemma 7.2] to the present context. Thus, adapting the proof of [3, Theorem 1.4] amounts to replacing each Brownian scaling in [3, Section 7] with stable scaling.
Proof of Theorem 1.3: Our proof follows closely the outline of [3, Section 8], to which the reader is referred for notation and details, where we take nowh(ǫ) =ǫβ|logǫ|andδn=ǫβnρ(ǫn)
as needed for applying the lower bound of (2.8). Forρn =ǫβn|logǫn|−3β and the i.i.d. random
variables
Yi(n) := 1A(n,i) h(ǫn)
Z 2iδn
(2i−1)δn
1{|X2
iδn−Xs|<ǫnbǫn}ds ,
where
A(n, i) = { sup
t∈(0,ρn)
|X2iδn+t−X2iδn| ≤ǫn|logǫn|
−2}
we combine [1, Proposition VIII.4] with (2.8) to provide the lower bound onP(Y(n) ≥a/(1−δ)) leading to [3, (8.2)] (see the derivation of (4.2) for a similar application).
Proof of Theorem 1.5: We follow the outline of [3, Section 9], relying on Lemma 3.1 to control the exponential moments instead of Girsanov’s theorem used in [3, Lemma 9.1].
To deal with the occupation measure of balls not centered at the origin take 0< α <1 and fix b= 1 +α >1. Fork∈(1,∞), let Γk={x:|x| ∈[k−1, k]}and for a >0,
Da:={x∈Γk
lim infǫ→0
µX
T(B(x, ǫ))
Set ηn = 2−n and δn = η1−b
Thus, by Borel-Cantelli, it follows that a.s. An is empty for all n≥m2(ω), implying that the
setsDa are a.s. empty for allT <∞. By [13, Lemma 5], a.s.
Thus, takingk↑ ∞ completes the proof of the right side of (1.9), subject only to (4.6).
Thus, forv=bη, by (4.7), (4.8) and Chebycheff’s inequality,
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